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| 19125 | If we define truth, we can eliminate it |
| Full Idea: If truth can be explicitly defined, it can be eliminated. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.3) | |||
| A reaction: That we could just say p corresponds to the facts, or p coheres with our accepted beliefs, or p is the aim of our enquiries, and never mention the word 'true'. Definition is a strategy for reduction or elimination. |
| 19128 | If a language cannot name all objects, then satisfaction must be used, instead of unary truth |
| Full Idea: If axioms are formulated for a language (such as set theory) that lacks names for all objects, then they require the use of a satisfaction relation rather than a unary truth predicate. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 3.3) | |||
| A reaction: I take it this is an important idea for understanding why Tarski developed his account of truth based on satisfaction. |
| 19120 | Semantic theories need a powerful metalanguage, typically including set theory |
| Full Idea: Semantic approaches to truth usually necessitate the use of a metalanguage that is more powerful than the object-language for which it provides a semantics. It is usually taken to include set theory. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1) | |||
| A reaction: This is a motivation for developing an axiomatic account of truth, that moves it into the object language. |
| 19127 | The T-sentences are deductively weak, and also not deductively conservative |
| Full Idea: Although the theory is materially adequate, Tarski thought that the T-sentences are deductively too weak. …Also it seems that the T-sentences are not conservative, because they prove in PA that 0=0 and ¬0=0 are different, so at least two objects exist. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 3.2) | |||
| A reaction: They are weak because they can't prove completeness. This idea give two reasons for looking for a better theory of truth. |
| 19124 | A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness |
| Full Idea: If a natural theory of truth is added to Peano Arithmetic, it is not necessary to add explicity global reflection principles to assert soundness, as the truth theory proves them. Truth theories thus prove soundess, and allows its expression. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.2) | |||
| A reaction: This seems like a big attraction of axiomatic theories of truth for students of metamathematics. |
| 19126 | If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new |
| Full Idea: If truth does not have any explanatory force, as some deflationists claim, the axioms of truth should not allow us to prove any new theorems that do not involve the truth predicate. That is, a deflationary axiomatisation of truth should be 'conservative'. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.3) | |||
| A reaction: So does truth have 'explanatory force'? These guys are interested in explaining theorems of arithmetic, but I'm more interested in real life. People do daft things because they have daft beliefs. Logic should be neutral, but truth has values? |
| 19129 | The FS axioms use classical logical, but are not fully consistent |
| Full Idea: It is a virtue of the Friedman-Sheard axiomatisation that it is thoroughly classical in its logic. Its drawback is that it is ω-inconsistent. That is, it proves &exists;x¬φ(x), but proves also φ(0), φ(1), φ(2), … | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.3) | |||
| A reaction: It seems the theory is complete (and presumably sound), yet not fully consistent. FS also proves the finite levels of Tarski's hierarchy, but not the transfinite levels. |
| 19130 | KF is formulated in classical logic, but describes non-classical truth, which allows truth-value gluts |
| Full Idea: KF is formulated in classical logic, but describes a non-classical notion of truth. It allow truth-value gluts, making some sentences (such as the Liar) both true and not-true. Some authors add an axiom ruling out such gluts. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.4) | |||
| A reaction: [summary, which I hope is correct! Stanford is not wholly clear] |
| 19121 | We can reduce properties to true formulas |
| Full Idea: One might say that 'x is a poor philosopher' is true of Tom instead of saying that Tom has the property of being a poor philosopher. We quantify over formulas instead of over definable properties, and thus reduce properties to truth. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.1) | |||
| A reaction: [compressed] This stuff is difficult (because the axioms are complex and hard to compare), but I am excited (yes!) about this idea. Their point is that you need a truth predicate within the object language for this, which disquotational truth forbids. |
| 19122 | Nominalists can reduce theories of properties or sets to harmless axiomatic truth theories |
| Full Idea: The reduction of second-order theories (of properties or sets) to axiomatic theories of truth is a form of reductive nominalism, replacing existence assumptions (e.g. comprehension axioms) by innocuous assumptions about the truth predicate. | |||
| From: Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.1) | |||
| A reaction: I'm currently thinking that axiomatic theories of truth are the most exciting development in contemporary philosophy. See Halbach and Horsten. |